Suppose F=⟨−y,x,z⟩ and S is the part of the sphere x2+y2+z2=25 below the plane z=4, oriented with the outward-pointing normal (so that the normal at (5,0,0) is i ). Compute the flux integral ∬Scurl F.dS using Stokes' theorem.
Use Stokes' theorem to evaluate ∬Scur∣F.dS where F=z2i−3xyj+x3y3k and S is the part of z=5−x2−y2 above the z=1. Assume that S is oriented upwards.
Use Stokes' theorem to evaluate ∫cF.dr where F=z2i+y2j+xk and C is the triangle with vertices (1,0,0),(0,1,0) and (0,0,1) with counter clockwise rotation.
Verify Stokes' Theorem for the field F=⟨x2,2x,z2⟩ on the ellipse S={(x,y,z):4x2+y2≤4,z=0}
Verify Stokes' Theorem for F=⟨y2,−x,5z⟩ and S is the paraboloid z=x2+y2 with the circle x2+y2=1 as its boundary.
Use Stokes' Theorem to calculate ∬(∇×F)⋅n^dS for F=⟨xz2,x3,cos(xz)⟩ where S is the part of the ellipsoid xx2+y2+3z2=1 below the xy-plane and n^ is the lower normal.
Use Stoke's Theorem to evaluate the line integral ∫F⋅drc where F is the vector F= (4ex2−y)i+(16sin(y2)+3x)j+(4y−2x−ez)k and C is the curve of intersection of the cylinder x2+y2=16 and the plane z=2x+4y and C is oriented in a counterclockwise direction when viewed from above.
Evaluate the line integral of F(x,y,z)=⟨xy,2z,3y⟩ over the curve C that is the intersection of the cylinder x2+y2=9 with the plane x+z=5.
Evaluate ∬(∇×F)⋅ndS where F(x,y,z)=⟨yz,xz,xy⟩ and S is the part of the sphere x2+y2+z2=4 that lies inside the cylinder x2+y2=1 and above the xy-plane.